$$U_{415}(x, y) = - \frac{\sqrt{4 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}}{2} + \frac{\sqrt{4 y + 1}}{4} + \frac{1}{4}$$

$$T_{415}(n, m, k) = \begin{cases}- \frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\0,\ \\0&\text{if n<k} ,\ \\- \frac{\left(-1\right)^{m - 1} k {\binom{k - n - 1}{n}} {\binom{- k + 2 m + 2 n - 1}{m - 1}}}{m}&\text{if m>0,n>k} \end{cases} $$